Prove the following two theorems i) A sequence in Rn can have at most one limit. Prove the following two theorems i) A sequence in Rn can have at most one limit. ii) If {xk}k in N is a sequence in Rn which converges to a and { xkj converges to a as j to infinite. xkj}j in N is a subsequence of {xk}k in N, then Solution If possible let a sequence n Rn have two limits a and b Then |Rn-a|N This gives |a-b|< delta which suggests that a=b Hence at most only one limit. ------------------------------------------------------------------------------------------- Rn converges to a in a sequence N. Then the subsequence term also converges..